Let $b$ be a $BMO$-function. It is well-known that the linear commutator $[b, T]$ of a Calderon-Zygmund operator $T$ does not, in general, map continuously $H^1(\mathbb R^n)$ into $L^1(\mathbb R^n)$. However, Perez showed that if $H^1(\mathbb R^n)$ is replaced by a suitable atomic subspace $\mathcal H^1_b(\mathbb R^n)$ then the commutator is continuous from $\mathcal H^1_b(\mathbb R^n)$ into $L^1(\mathbb R^n)$. In this paper, we find the largest subspace $H^1_b(\mathbb R^n)$ such that all commutators of Calderon-Zygmund operators are continuous from $H^1_b(\mathbb R^n)$ into $L^1(\mathbb R^n)$. Some equivalent characterizations of $H^1_b(\mathbb R^n)$ are also given. We also study the commutators $[b,T]$ for $T$ in a class $\mathcal K$ of sublinear operators containing almost all important operators in harmonic analysis. When $T$ is linear, we prove that there exists a bilinear operators $\mathfrak R= \mathfrak R_T$ mapping continuously $H^1(\mathbb R^n)\times BMO(\mathbb R^n)$ into $L^1(\mathbb R^n)$ such that for all $(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)$, we have \begin{equation}\label{abstract 1} [b,T](f)= \mathfrak R(f,b) + T(\mathfrak S(f,b)), \end{equation} where $\mathfrak S$ is a bounded bilinear operator from $H^1(\mathbb R^n)\times BMO(\mathbb R^n)$ into $L^1(\mathbb R^n)$ which does not depend on $T$. In the particular case of $T$ a Calderon-Zygmund operator satisfying $T1=T^*1=0$ and $b$ in $BMO^{\rm log}(\mathbb R^n)$-- the generalized $\BMO$ type space that has been introduced by Nakai and Yabuta to characterize multipliers of $\BMO(\bR^n)$ --we prove that the commutator $[b,T]$ maps continuously $H^1_b(\mathbb R^n)$ into $h^1(\mathbb R^n)$. Also, if $b$ is in $BMO(\mathbb R^n)$ and $T^*1 = T^*b = 0$, then the commutator $[b, T]$ maps continuously $H^1_b (\mathbb R^n)$ into $H^1(\mathbb R^n)$. When $T$ is sublinear, we prove that there exists a bounded subbilinear operator $\mathfrak R= \mathfrak R_T: H^1(\mathbb R^n)\times BMO(\mathbb R^n)\to L^1(\mathbb R^n)$ such that for all $(f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n)$, we have \begin{equation}\label{abstract 2} |T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|. \end{equation} The bilinear decomposition (\ref{abstract 1}) and the subbilinear decomposition (\ref{abstract 2}) allow us to give a general overview of all known weak and strong $L^1$-estimates.

https://hal.archives-ouvertes.fr/hal-00589824/document

Bilinear decompositions and commutators of singular integral operators

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Let μ be a positive Radon measure on ${\mathbb{R}}^d$ which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Crn for all $x \in {\mathbb{R}}^d$ , r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some Hormander-type condition, and assume that it is bounded on L2(μ). We then establish its boundedness, respectively, from the Lebesgue space L1(μ) to the weak Lebesgue space L1,∞(μ), from the Hardy space H1(μ) to L1(μ) and from the Lebesgue space L∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space Lp(μ) with p ∈ (1,∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(μ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger Hormander-type condition, respectively, from Lp(μ) with p ∈ (1,∞) to itself, from the space L log L(μ) to L1,∞(μ) and from H1(μ) to L1,∞(μ). Some of the results are also new even for the classical Marcinkiewicz integral.

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We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given.

Characterization of boundedness of some commutators of maximal functions in terms of Lipschitz spaces

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$$1 < p_ - : = \mathop {es\sin fp(x)}\limits_{x \in \mathbb{R}^n } andp_ + : = \mathop {es\operatorname{s} \sup p(x) < \infty }\limits_{x \in \mathbb{R}^n } ,$$

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/pdf/commutators-of-the-fractional-maximal-function-on-variable-189mkys2bx.pdf

Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let $$0\le \alpha <n$$
 , $$[b,M_{\alpha }]$$
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/pdf/commutators-for-the-maximal-functions-on-lebesgue-spaces-49our7cy62.pdf

Commutators for the maximal functions on Lebesgue spaces with variable exponent

Let (0 1) boundedness, the weak (L1, Ln/(n–α–β)) boundedness and the (Lp, Ḟβ, ∞p) boundedness of are discussed, when DγA belongs to the Lipschitz function spaces.

Lipschitz estimates for generalized commutators of fractional integrals with rough kernel

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